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Damped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation d 2 y d t 2 + 2 λ d y d t + ω 2 y = 0 The zeros of its characteristic equation are m 1 = − λ + λ 2 − ω 2 and m 1 = − λ − λ 2 − ω 2 For λ 2 − ω 2 > 0 , the system is overdamped; λ 2 − ω 2 = 0 , it is critically damped; and for λ 2 − ω 2 < 0 , it is underdamped. Determine whether the differential equation represents an overdamped, critically damped, or underdamped system. Find the particular solution that satisfies the initial conditions. Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system. d 2 y d t 2 + 8 d y d t + 16 y = 0 y ( 0 ) = 1 and y ′ ( 0 ) = 1

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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 11PS
Textbook Problem
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Damped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation

d 2 y d t 2 + 2 λ d y d t + ω 2 y = 0

The zeros of its characteristic equation are

m 1 = λ + λ 2 ω 2

and

m 1 = λ λ 2 ω 2

For λ 2 ω 2 > 0 , the system is overdamped; λ 2 ω 2 = 0 , it is critically damped; and for λ 2 ω 2 < 0 , it is underdamped.

Determine whether the differential equation represents an overdamped, critically damped, or underdamped system.

Find the particular solution that satisfies the initial conditions.

Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system.

d 2 y d t 2 + 8 d y d t + 16 y = 0

y ( 0 ) = 1 and y ( 0 ) = 1

(a)

To determine

Whether the differential equation d2ydt2+8dydt+16y=0 represents an overdamped, critically damped, or under damped system.

Explanation of Solution

Given information: The zeros of the characteristic equation of the differential equation d2ydt2+2λdydt+ω2y=0 are m1=λ+λ2ω2 and m1=λλ2ω2.

For λ2ω2>0, the system is over damped; for λ2ω2=0, it is critically damped and for λ2ω2<0, it is underdamped

(b)

To determine

The particular solution of the differential equation d2ydt2+8dydt+16y=0 that satisfies the initial conditions y(0)=1andy(0)=1.

(c)

To determine

To graph: The particular solution y=(1+5t)e4t using a graphing utility and explain how the graph illustrates the type of damping in part (a).

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Chapter 16 Solutions

Multivariable Calculus
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